The circle method from [1]} uses of smooth weight functions \(w:\mathbb {R}^4\rightarrow \mathbb {R}_{\geqslant 0}\) , which we will take to approximate the characteristic function on \([-1,1]^4\) . We introduce a smoothly weighted version of \(N_{\mathbf {a}}(B)\)  given by
\(N_{w,\mathbf {a}}(B)=\sum _{\begin{array}{c}\mathbf {x}\in (\mathbb {Z}_{\ne 0 })^4\\F_{\mathbf {a}}(\mathbf {x}) = 0\end{array}}w(P_1^{-1}x_1, \ldots , P_4^{-1}x_4).\) 
